safelanes.c

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/*
 * See Licensing and Copyright notice in naev.h
 */
#include <math.h>
 
#if HAVE_SUITESPARSE_CHOLMOD_H
#include <suitesparse/cholmod.h>
#else /* HAVE_SUITESPARSE_CHOLMOD_H */
#include <cholmod.h>
#endif /* HAVE_SUITESPARSE_CHOLMOD_H */
 
#if HAVE_OPENBLAS_CBLAS_H
#   include <openblas/cblas.h>
#elif HAVE_CBLAS_OPENBLAS_H
#   include <cblas_openblas.h>
#elif HAVE_CBLAS_HYPHEN_OPENBLAS_H
#   include <cblas-openblas.h>
#elif HAVE_ACCELERATE_ACCELERATE_H
#   include <Accelerate/Accelerate.h>
#elif HAVE_CBLAS_H
#   include <cblas.h>
#elif HAVE_F77BLAS_H
#   include <f77blas.h>
#   define I_LOVE_FORTRAN 1
#elif HAVE_OPENBLAS_F77BLAS_H
#   include <openblas/f77blas.h>
#   define I_LOVE_FORTRAN 1
#endif
 
#include "naev.h"
#include "safelanes.h"
 
#include "array.h"
#include "conf.h"
#include "log.h"
#include "union_find.h"
 
/*
 * Global parameters.
 */
static const double ALPHA            = 9.;       
static const double LAMBDA           = 2e10;     
static const double JUMP_CONDUCTIVITY= 0.001;    
static const double MIN_ANGLE        = M_PI/18.; 
enum {
   STORAGE_MODE_LOWER_TRIANGULAR_PART= -1,       
   STORAGE_MODE_UNSYMMETRIC          = 0,        
   STORAGE_MODE_UPPER_TRIANGULAR_PART= +1,       
   SORTED                            = 1,        
   PACKED                            = 1,        
   MODE_NUMERICAL                    = 1,        
};
 
/*
 * Types.
 */
typedef enum VertexType_ {VERTEX_SPOB, VERTEX_JUMP} VertexType;
 
typedef struct Vertex_ {
   int system;      
   VertexType type; 
   int index;       
} Vertex;
 
typedef int Edge[2];
 
typedef struct Faction_ {
   int id;                          
   double lane_length_per_presence; 
   double lane_base_cost;           
} Faction;
 
typedef uint32_t FactionMask;
static const FactionMask MASK_0 = 0, MASK_1 = 1;
 
/*
 * Global state.
 */
static cholmod_common C;        
static Vertex *vertex_stack;    
static FactionMask *vertex_fmask;
static int *sys_to_first_vertex;
static Edge *edge_stack;        
static int *sys_to_first_edge;  
static Faction *faction_stack;  
static int *lane_faction;       
static FactionMask *lane_fmask; 
static double **presence_budget;
static int *tmp_spob_indices; 
static Edge *tmp_jump_edges;    
static double *tmp_edge_conduct;
static int *tmp_anchor_vertices;
static UnionFind tmp_sys_uf;    
static cholmod_triplet *stiff;  
static cholmod_sparse *QtQ;     
static cholmod_dense *ftilde;   
static cholmod_dense *utilde;   
static cholmod_dense **PPl;     
static double* cmp_key_ref;     
static int safelanes_calculated_once = 0; 
/*
 * Prototypes.
 */
static int safelanes_buildOneTurn( int iters_done );
static int safelanes_activateByGradient( cholmod_dense* Lambda_tilde, int iters_done );
static void safelanes_initStacks (void);
static void safelanes_initStacks_edge (void);
static void safelanes_initStacks_faction (void);
static void safelanes_initStacks_vertex (void);
static void safelanes_initStacks_anchor (void);
static void safelanes_initOptimizer (void);
static void safelanes_destroyOptimizer (void);
static void safelanes_destroyStacks (void);
static void safelanes_destroyTmp (void);
static void safelanes_initStiff (void);
static double safelanes_initialConductivity ( int ei );
static void safelanes_updateConductivity ( int ei_activated );
static void safelanes_initQtQ (void);
static void safelanes_initFTilde (void);
static void safelanes_initPPl (void);
static int safelanes_triangleTooFlat( const vec2* m, const vec2* n, const vec2* p, double lmn );
static int vertex_faction( int vi );
static const vec2* vertex_pos( int vi );
static inline int FACTION_ID_TO_INDEX( int id );
static inline FactionMask MASK_ANY_FACTION();
static inline FactionMask MASK_ONE_FACTION( int id );
static inline FactionMask MASK_COMPROMISE( int id1, int id2 );
static int cmp_key( const void* p1, const void* p2 );
static inline void triplet_entry( cholmod_triplet* m, int i, int j, double v );
static cholmod_dense* safelanes_sliceByPresence( cholmod_dense* m, double* sysPresence );
static cholmod_dense* ncholmod_ddmult( cholmod_dense* A, int transA, cholmod_dense* B );
static double safelanes_row_dot_row( cholmod_dense* A, cholmod_dense* B, int i, int j );
 
static inline void array_push_back_edge( Edge **a, int v0, int v1 )
{
   int *vs = array_grow( a );
   vs[0] = v0;
   vs[1] = v1;
}
 
void safelanes_init (void)
{
   cholmod_start( &C );
   /* Ideally we would want to recalculate here, but since we load the first
    * save and try to use unidiffs there, we instead defer the safe lane
    * computation to only if necessary after loading save unidiffs. */
   /* safelanes_recalculate(); */
}
 
void safelanes_destroy (void)
{
   safelanes_destroyOptimizer();
   safelanes_destroyStacks();
   cholmod_finish( &C );
}
 
SafeLane* safelanes_get( int faction, int standing, const StarSystem* system )
{
   SafeLane *out = array_create( SafeLane );
 
   for (int i=sys_to_first_edge[system->id]; i<sys_to_first_edge[1+system->id]; i++) {
      SafeLane *l;
      const Vertex *v[2];
      int lf = lane_faction[i];
 
      /* No lane on edge. */
      if (lf <= 0)
         continue;
 
      /* Filter by standing. */
      if (faction >= 0) {
         if (standing==0) {
            /* Only match exact faction. */
            if (lf != faction)
               continue;
         }
         else {
            /* Try to do more advanced matching. */
            int fe = areEnemies(faction,lf);
            int fa = areAllies(faction,lf);
            int skip = 1;
            if ((standing & SAFELANES_FRIENDLY) && fa)
               skip = 0;
            if ((standing & SAFELANES_NEUTRAL) && !fe)
               skip = 0;
            if ((standing & SAFELANES_HOSTILE) && fe)
               skip = 0;
            if (skip)
               continue;
         }
      }
 
      for (int j=0; j<2; j++)
         v[j] = &vertex_stack[edge_stack[i][j]];
 
      l = &array_grow( &out );
      l->faction = lane_faction[i];
      for (int j=0; j<2; j++) {
         switch (v[j]->type) {
            case VERTEX_SPOB:
               l->point_type[j]   = SAFELANE_LOC_SPOB;
               l->point_id[j]     = system->spobs[v[j]->index]->id;
               break;
            case VERTEX_JUMP:
               l->point_type[j]   = SAFELANE_LOC_DEST_SYS;
               l->point_id[j]     = system->jumps[v[j]->index].targetid;
               break;
            default:
               ERR( _("Safe-lane vertex type is invalid.") );
         }
      }
   }
   return out;
}
 
void safelanes_recalculate (void)
{
   Uint32 time = SDL_GetTicks();
 
   /* Don't recompute on exit. */
   if (naev_isQuit())
      return;
 
   safelanes_initStacks();
   safelanes_initOptimizer();
   for (int iters_done=0; safelanes_buildOneTurn(iters_done) > 0; iters_done++)
      ;
   safelanes_destroyOptimizer();
   /* Stacks remain available for queries. */
   time = SDL_GetTicks() - time;
   if (conf.devmode)
      DEBUG( n_("Charted safe lanes for %d object in %.3f s", "Charted safe lanes for %d objects in %.3f s", array_size(vertex_stack)), array_size(vertex_stack), time/1000. );
 
   safelanes_calculated_once = 1;
}
 
int safelanes_calculated (void)
{
   return safelanes_calculated_once;
}
 
static void safelanes_initOptimizer (void)
{
   safelanes_initStiff();
   safelanes_initQtQ();
   safelanes_initFTilde();
   safelanes_initPPl();
   safelanes_destroyTmp();
}
 
static void safelanes_destroyOptimizer (void)
{
   for (int i=0; i<array_size(PPl); i++)
      cholmod_free_dense( &PPl[i], &C );
   array_free( PPl );
   PPl = NULL;
   cholmod_free_dense( &utilde, &C ); /* CAUTION: if we instead save it, ensure it's updated after the final activateByGradient. */
   cholmod_free_dense( &ftilde, &C );
   cholmod_free_sparse( &QtQ, &C );
   cholmod_free_triplet( &stiff, &C );
}
 
static int safelanes_buildOneTurn( int iters_done )
{
   cholmod_sparse *stiff_s;
   cholmod_factor *stiff_f;
   cholmod_dense *_QtQutilde, *Lambda_tilde, *Y_workspace, *E_workspace;
   int turns_next_time;
   double zero[] = {0, 0}, neg_1[] = {-1, 0};
 
   Y_workspace = E_workspace = Lambda_tilde = NULL;
   stiff_s = cholmod_triplet_to_sparse( stiff, 0, &C );
   stiff_f = cholmod_analyze( stiff_s, &C );
   cholmod_factorize( stiff_s, stiff_f, &C );
   cholmod_solve2( CHOLMOD_A, stiff_f, ftilde, NULL, &utilde, NULL, &Y_workspace, &E_workspace, &C );
   _QtQutilde = cholmod_zeros( utilde->nrow, utilde->ncol, CHOLMOD_REAL, &C );
   cholmod_sdmult( QtQ, 0, neg_1, zero, utilde, _QtQutilde, &C );
   cholmod_solve2( CHOLMOD_A, stiff_f, _QtQutilde, NULL, &Lambda_tilde, NULL, &Y_workspace, &E_workspace, &C );
   cholmod_free_dense( &_QtQutilde, &C );
   cholmod_free_dense( &Y_workspace, &C );
   cholmod_free_dense( &E_workspace, &C );
   cholmod_free_factor( &stiff_f, &C );
   cholmod_free_sparse( &stiff_s, &C );
   turns_next_time = safelanes_activateByGradient( Lambda_tilde, iters_done );
   cholmod_free_dense( &Lambda_tilde, &C );
 
   return turns_next_time;
}
 
static void safelanes_initStacks (void)
{
   safelanes_destroyStacks();
   safelanes_initStacks_faction(); /* Dependency for vertex. */
   safelanes_initStacks_vertex(); /* Dependency for edge. */
   safelanes_initStacks_edge();
   safelanes_initStacks_anchor();
}
 
static void safelanes_initStacks_vertex (void)
{
   const StarSystem *systems_stack = system_getAll();
 
   vertex_stack = array_create( Vertex );
   sys_to_first_vertex = array_create( int );
   array_push_back( &sys_to_first_vertex, 0 );
   tmp_spob_indices = array_create( int );
   tmp_jump_edges = array_create( Edge );
   for (int system=0; system<array_size(systems_stack); system++) {
      const StarSystem *sys = &systems_stack[system];
      if (sys_isFlag( sys, SYSTEM_NOLANES )) {
         array_push_back( &sys_to_first_vertex, array_size(vertex_stack) );
         continue;
      }
 
      for (int i=0; i<array_size(sys->spobs); i++) {
         const Spob *p = sys->spobs[i];
         if (spob_isFlag( p, SPOB_NOLANES ))
            continue;
         if (p->presence.base!=0. || p->presence.bonus!=0.) {
            Vertex v = {.system = system, .type = VERTEX_SPOB, .index = i};
            array_push_back( &tmp_spob_indices, array_size(vertex_stack) );
            array_push_back( &vertex_stack, v );
         }
      }
 
      for (int i=0; i<array_size(sys->jumps); i++) {
         const JumpPoint *jp = &sys->jumps[i];
         if (jp_isFlag( jp, JP_HIDDEN | JP_EXITONLY | JP_NOLANES ))
            continue;
         Vertex v = {.system = system, .type = VERTEX_JUMP, .index = i};
         array_push_back( &vertex_stack, v );
         if (jp->targetid < system && jp->returnJump != NULL)
            for (int j=sys_to_first_vertex[jp->targetid]; j < sys_to_first_vertex[1+jp->targetid]; j++)
               if (vertex_stack[j].type == VERTEX_JUMP && jp->returnJump == &jp->target->jumps[vertex_stack[j].index]) {
                  array_push_back_edge( &tmp_jump_edges, array_size(vertex_stack)-1, j );
                  break;
               }
      }
 
      array_push_back( &sys_to_first_vertex, array_size(vertex_stack) );
   }
   //array_shrink( &vertex_stack );
   //array_shrink( &sys_to_first_vertex );
 
   vertex_fmask = calloc( array_size(vertex_stack), sizeof(FactionMask) );
}
 
static void safelanes_initStacks_edge (void)
{
   const StarSystem *systems_stack = system_getAll();
 
   edge_stack = array_create( Edge );
   sys_to_first_edge = array_create( int );
   array_push_back( &sys_to_first_edge, 0 );
   lane_fmask = array_create( FactionMask );
   tmp_edge_conduct = array_create( double );
   for (int system=0; system<array_size(systems_stack); system++) {
      for (int i=sys_to_first_vertex[system]; i < sys_to_first_vertex[1+system]; i++) {
         const vec2 *pi = vertex_pos( i );
         for (int j=sys_to_first_vertex[system]; j < i; j++) {
            const vec2 *pj = vertex_pos( j );
            double lij = vec2_dist( pi, pj );
            int has_approx_midpoint = 0;
            for (int k=sys_to_first_vertex[system]; k < sys_to_first_vertex[1+system]; k++)
               if (k!=i && k!=j && safelanes_triangleTooFlat( pi, pj, vertex_pos( k ), lij )) {
                  has_approx_midpoint = 1;
                  break;
               }
            if (has_approx_midpoint)
               continue;  /* The edge from i to j is disallowed in favor of a path through k. */
            array_push_back_edge( &edge_stack, j, i );
            array_push_back( &lane_fmask, MASK_COMPROMISE( vertex_faction( i ), vertex_faction( j ) ) );
            array_push_back( &tmp_edge_conduct, 1/lij );
         }
      }
      array_push_back( &sys_to_first_edge, array_size(edge_stack) );
   }
   array_shrink( &edge_stack );
   array_shrink( &sys_to_first_edge );
 
   lane_faction = array_create_size( int, array_size(edge_stack) );
   array_resize( &lane_faction, array_size(edge_stack) );
   memset( lane_faction, 0, array_size(lane_faction)*sizeof(lane_faction[0]) );
}
 
static void safelanes_initStacks_faction (void)
{
   int *faction_all;
   const StarSystem *systems_stack;
 
   faction_stack = array_create( Faction );
   faction_all = faction_getAllVisible();
   for (int fi=0; fi<array_size(faction_all); fi++) {
      int f = faction_all[fi];
      Faction rec = {.id = f, .lane_length_per_presence = faction_lane_length_per_presence(f), .lane_base_cost = faction_lane_base_cost(f)};
      if (rec.lane_length_per_presence > 0.)
         array_push_back( &faction_stack, rec );
   }
   array_free( faction_all );
   array_shrink( &faction_stack );
   assert( "FactionMask size is sufficient" && (size_t)array_size(faction_stack) <= 8*sizeof(FactionMask) );
 
   presence_budget = array_create_size( double*, array_size(faction_stack) );
   systems_stack = system_getAll();
   for (int fi=0; fi<array_size(faction_stack); fi++) {
      array_push_back( &presence_budget, array_create_size( double, array_size(systems_stack) ) );
      for (int s=0; s<array_size(systems_stack); s++) {
         const StarSystem *sys = &systems_stack[s];
         double budget = system_getPresence( sys, faction_stack[fi].id );
         array_push_back( &presence_budget[fi], budget );
      }
   }
}
 
static void safelanes_initStacks_anchor (void)
{
   int *anchor_systems;
   int nsys = array_size(sys_to_first_vertex) - 1;
   unionfind_init( &tmp_sys_uf, nsys );
   for (int i=0; i<array_size(tmp_jump_edges); i++)
      unionfind_union( &tmp_sys_uf, vertex_stack[tmp_jump_edges[i][0]].system, vertex_stack[tmp_jump_edges[i][1]].system );
   anchor_systems = unionfind_findall( &tmp_sys_uf );
   tmp_anchor_vertices = array_create_size( int, array_size(anchor_systems) );
 
   /* Add an anchor vertex per system, but only if there actually is a vertex in the system. */
   for (int i=0; i<array_size(anchor_systems); i++)
      if (sys_to_first_vertex[anchor_systems[i]] < sys_to_first_vertex[1+anchor_systems[i]])
         array_push_back( &tmp_anchor_vertices, sys_to_first_vertex[anchor_systems[i]] );
   array_free( anchor_systems );
}
 
static void safelanes_destroyStacks (void)
{
   safelanes_destroyTmp();
   array_free( vertex_stack );
   vertex_stack = NULL;
   free( vertex_fmask );
   vertex_fmask = NULL;
   array_free( sys_to_first_vertex );
   sys_to_first_vertex = NULL;
   array_free( edge_stack );
   edge_stack = NULL;
   array_free( sys_to_first_edge );
   sys_to_first_edge = NULL;
   for (int i=0; i<array_size(presence_budget); i++)
      array_free( presence_budget[i] );
   array_free( presence_budget );
   presence_budget = NULL;
   array_free( faction_stack );
   faction_stack = NULL;
   array_free( lane_faction );
   lane_faction = NULL;
   array_free( lane_fmask );
   lane_fmask = NULL;
}
 
static void safelanes_destroyTmp (void)
{
   unionfind_free( &tmp_sys_uf );
   array_free( tmp_spob_indices );
   tmp_spob_indices = NULL;
   array_free( tmp_jump_edges );
   tmp_jump_edges = NULL;
   array_free( tmp_edge_conduct );
   tmp_edge_conduct = NULL;
   array_free( tmp_anchor_vertices );
   tmp_anchor_vertices = NULL;
}
 
static void safelanes_initStiff (void)
{
   int nnz, v;
   double max_conductivity;
 
   cholmod_free_triplet( &stiff, &C );
   v = array_size(vertex_stack);
   nnz = 3*(array_size(edge_stack)+array_size(tmp_jump_edges)) + array_size(tmp_anchor_vertices);
   stiff = cholmod_allocate_triplet( v, v, nnz, STORAGE_MODE_UPPER_TRIANGULAR_PART, CHOLMOD_REAL, &C );
   /* Populate triplets: internal edges (ii ij jj), implicit jump connections (ditto), anchor conditions. */
   for (int i=0; i<array_size(edge_stack); i++) {
      triplet_entry( stiff, edge_stack[i][0], edge_stack[i][0], +tmp_edge_conduct[i] );
      triplet_entry( stiff, edge_stack[i][0], edge_stack[i][1], -tmp_edge_conduct[i] );
      triplet_entry( stiff, edge_stack[i][1], edge_stack[i][1], +tmp_edge_conduct[i] );
   }
   for (int i=0; i<array_size(tmp_jump_edges); i++) {
      triplet_entry( stiff, tmp_jump_edges[i][0], tmp_jump_edges[i][0], +JUMP_CONDUCTIVITY );
      triplet_entry( stiff, tmp_jump_edges[i][0], tmp_jump_edges[i][1], -JUMP_CONDUCTIVITY );
      triplet_entry( stiff, tmp_jump_edges[i][1], tmp_jump_edges[i][1], +JUMP_CONDUCTIVITY );
   }
   /* Add a Robin boundary condition, using the max conductivity (after activation) for spectral reasons. */
   max_conductivity = JUMP_CONDUCTIVITY/(1+ALPHA);
   for (int i=0; i<array_size(edge_stack); i++)
      max_conductivity = MAX( max_conductivity, tmp_edge_conduct[i] );
   max_conductivity = MAX( JUMP_CONDUCTIVITY, (1+ALPHA)*max_conductivity ); /* Activation scales entries by 1+ALPHA later. */
   for (int i=0; i<array_size(tmp_anchor_vertices); i++)
      triplet_entry( stiff, tmp_anchor_vertices[i], tmp_anchor_vertices[i], max_conductivity );
#if DEBUGGING
   assert( stiff->nnz == stiff->nzmax );
   assert( cholmod_check_triplet( stiff, &C) );
#endif /* DEBUGGING */
}
 
static double safelanes_initialConductivity ( int ei )
{
   double *sv = stiff->x;
   return lane_faction[ei] ? sv[3*ei]/(1+ALPHA) : sv[3*ei];
}
 
static void safelanes_updateConductivity ( int ei_activated )
{
   double *sv = stiff->x;
   for (int i=3*ei_activated; i<3*(ei_activated+1); i++)
      sv[i] *= 1+ALPHA;
}
 
static void safelanes_initQtQ (void)
{
   cholmod_sparse *Q;
 
   cholmod_free_sparse( &QtQ, &C );
   /* Form Q, the edge-vertex projection where (Dirac notation) Q |edge> = |edge[0]> - |edge[1]>. It has a +1 and -1 per column. */
   Q = cholmod_allocate_sparse( array_size(vertex_stack), array_size(edge_stack), 2*array_size(edge_stack),
         SORTED, PACKED, STORAGE_MODE_UNSYMMETRIC, CHOLMOD_REAL, &C );
   ((int*)Q->p)[0] = 0;
   for (int i=0; i<array_size(edge_stack); i++) {
      ((int*)Q->p)[i+1] = 2*(i+1);
      ((int*)Q->i)[2*i+0] = edge_stack[i][0];
      ((int*)Q->i)[2*i+1] = edge_stack[i][1];
      ((double*)Q->x)[2*i+0] = +1;
      ((double*)Q->x)[2*i+1] = -1;
   }
#if DEBUGGING
   assert( cholmod_check_sparse( Q, &C ) );
#endif /* DEBUGGING */
   QtQ = cholmod_aat( Q, NULL, 0, MODE_NUMERICAL, &C );
   cholmod_free_sparse( &Q, &C );
}
 
static void safelanes_initFTilde (void)
{
   cholmod_sparse *eye = cholmod_speye( array_size(vertex_stack), array_size(vertex_stack), CHOLMOD_REAL, &C );
   cholmod_sparse *sp = cholmod_submatrix( eye, NULL, -1, tmp_spob_indices, array_size(tmp_spob_indices), 1, SORTED, &C );
   cholmod_free_dense( &ftilde, &C );
   ftilde = cholmod_sparse_to_dense( sp, &C );
   cholmod_free_sparse( &sp, &C );
   cholmod_free_sparse( &eye, &C );
}
 
static void safelanes_initPPl (void)
{
   int *component;
   int np = array_size(tmp_spob_indices);
 
   for (int fi=0; fi<array_size(PPl); fi++)
      cholmod_free_dense( &PPl[fi], &C );
   array_free( PPl );
   PPl = array_create_size( cholmod_dense*, array_size(faction_stack) );
   for (int fi=0; fi<array_size(faction_stack); fi++)
      array_push_back( &PPl, cholmod_zeros( np, np, CHOLMOD_REAL, &C ) );
 
   /* Form P, the pair-vertex projection where (Dirac notation) P |pair(i,j)> = |i> - |j>. It has a +1 and -1 per column. */
   /* At least, pretend we did. We want (PD)(PD)*, where D is a diagonal matrix whose pair(i,j) are these presence sums: */
 
   component = calloc( np, sizeof(int) );
   for (int i=0; i<np; i++)
      component[i] = unionfind_find( &tmp_sys_uf, vertex_stack[tmp_spob_indices[i]].system );
 
   for (int i=0; i<np; i++) {
      double *Di;
      int sys = vertex_stack[tmp_spob_indices[i]].system;
      Spob *pnt = system_getIndex( sys )->spobs[vertex_stack[tmp_spob_indices[i]].index];
      double pres = pnt->presence.base + pnt->presence.bonus; /* TODO distinguish between base and bonus? */
      int fi = FACTION_ID_TO_INDEX( pnt->presence.faction );
      if (fi < 0)
         continue;
      Di = PPl[fi]->x;
      for (int j=0; j<i; j++)
         if (component[i] == component[j])
            Di[np*i+j] += pres;
      for (int j=i+1; j<np; j++)
         if (component[i] == component[j])
            Di[np*j+i] += pres;
   }
 
   /* At this point, PPl[fi]->x[np*i+j] holds the pair(i,j) entry of D. */
   for (int fi=0; fi<array_size(faction_stack); fi++)
      for (int i=0; i<np; i++)
         for (int j=0; j<i; j++) {
            double d = ((double*)PPl[fi]->x)[np*i+j];
            d *= d;
            ((double*)PPl[fi]->x)[np*i+j] = -d;
            ((double*)PPl[fi]->x)[np*j+i] = -d;
            ((double*)PPl[fi]->x)[np*i+i] += d;
            ((double*)PPl[fi]->x)[np*j+j] += d;
         }
 
   free( component );
}
 
static int safelanes_activateByGradient( cholmod_dense* Lambda_tilde, int iters_done )
{
   int *facind_opts, *edgeind_opts, turns_next_time;
   double *facind_vals, Linv;
   cholmod_dense **lal; 
   size_t *lal_bases, lal_base; 
   lal = calloc( array_size(faction_stack), sizeof(cholmod_dense*) );
   lal_bases = calloc( array_size(faction_stack), sizeof(size_t) );
   edgeind_opts = array_create( int );
   facind_opts = array_create_size( int, array_size(faction_stack) );
   facind_vals = array_create_size( double, array_size(faction_stack) );
   for (int fi=0; fi<array_size(faction_stack); fi++) {
      array_push_back( &facind_opts, fi );
      array_push_back( &facind_vals, 0 );
   }
   turns_next_time = 0;
 
   for (int si=0; si<array_size(sys_to_first_vertex)-1; si++) {
      /* Factions with most presence here choose first. */
      StarSystem *sys = system_getIndex( si );
      for (int fi=0; fi<array_size(faction_stack); fi++)
         facind_vals[fi] = -system_getPresence( sys, faction_stack[fi].id ); /* FIXME: Is this better, or presence_budget? */
      cmp_key_ref = facind_vals;
      qsort( facind_opts, array_size(faction_stack), sizeof(int), cmp_key );
 
      for (int fii=0; fii<array_size(faction_stack); fii++) {
         int ei_best;
         double cost_best, cost_cheapest_other;
         int fi = facind_opts[fii];
         size_t sys_base = sys_to_first_vertex[si];
         double score_best = 0.; /* Negative scores get ignored. */
 
         /* Get the base index to use for this system. Save the value we expect to be the next iteration's base index.
          * The current system's rows are in the lal[fi] matrix if there's presence at the time we slice it.
          * What we know is whether there's presence *now*. We can use that as a proxy and fix lal_bases[fi] if we
          * deplete this presence before constructing lal[fi]. This is tricky, so there are assertions below,
          * which can warn us if we fuck this up. */
         lal_base = lal_bases[fi];
         if (presence_budget[fi][si] <= 0.)
            continue;
         /* We "should" find these DoF's interesting if/when we slice, and will unless we deplete this presence first. */
         lal_bases[fi] += sys_to_first_vertex[1+si] - sys_to_first_vertex[si];
 
         array_resize( &edgeind_opts, 0 );
         for (int ei=sys_to_first_edge[si]; ei<sys_to_first_edge[1+si]; ei++) {
            int sis = edge_stack[ei][0];
            int sjs = edge_stack[ei][1];
            int disconnecting = iters_done && !(vertex_fmask[sis] & (MASK_1<<fi)) && !(vertex_fmask[sjs] & (MASK_1<<fi));
            double cost = 1. / safelanes_initialConductivity(ei) / faction_stack[fi].lane_length_per_presence + faction_stack[fi].lane_base_cost;
            if (!lane_faction[ei]
                && !disconnecting
                && presence_budget[fi][si] >= cost
                && (lane_fmask[ei] & (MASK_1<<fi)))
               array_push_back( &edgeind_opts, ei );
         }
 
         if (array_size(edgeind_opts) == 0) {
            presence_budget[fi][si] = 0.;  /* Nothing to build here! Tell ourselves to stop trying. */
            if (lal[fi] == NULL)
               lal_bases[fi] -= sys_to_first_vertex[1+si] - sys_to_first_vertex[si];
            continue;
         }
 
         ei_best = edgeind_opts[0];
         cost_best = 1. / safelanes_initialConductivity(ei_best) / faction_stack[fi].lane_length_per_presence + faction_stack[fi].lane_base_cost;
         cost_cheapest_other = +HUGE_VAL;
         if (array_size(edgeind_opts) > 0) {
            /* There's an actual choice. Search for the best option. Lower is better. */
            for (int eii=0; eii<array_size(edgeind_opts); eii++) {
               int ei = edgeind_opts[eii];
               int sis = edge_stack[ei][0];
               int sjs = edge_stack[ei][1];
               double score = 0.;
               double cost;
 
               if (lal[fi] == NULL) { /* Is it time to evaluate the lazily-calculated matrix? */
                  cholmod_dense *lamt = safelanes_sliceByPresence( Lambda_tilde, presence_budget[fi] );
                  lal[fi] = ncholmod_ddmult( lamt, 0, PPl[fi] );
                  cholmod_free_dense( &lamt, &C );
               }
 
               /* Evaluate (LUTll[0,0] + LUTll[1,1] - LUTll[0,1] - LUTll[1,0]), */
               /* where    LUTll = np.dot( lal[[sis,sjs],:] , utilde[[sis,sjs],:].T ) */
               score += safelanes_row_dot_row( utilde, lal[fi], sis, sis - sys_base + lal_base );
               score += safelanes_row_dot_row( utilde, lal[fi], sjs, sjs - sys_base + lal_base );
               score -= safelanes_row_dot_row( utilde, lal[fi], sjs, sis - sys_base + lal_base );
               score -= safelanes_row_dot_row( utilde, lal[fi], sis, sjs - sys_base + lal_base );
               Linv = safelanes_initialConductivity(ei);
               score *= ALPHA * Linv * Linv;
               score += LAMBDA;
 
               cost = 1. / safelanes_initialConductivity(ei) / faction_stack[fi].lane_length_per_presence + faction_stack[fi].lane_base_cost;
               if (score < score_best) {
                  ei_best = ei;
                  score_best = score;
                  cost_cheapest_other = MIN( cost_cheapest_other, cost_best );
                  cost_best = cost;
               }
               else
                  cost_cheapest_other = MIN( cost_cheapest_other, cost );
            }
         }
 
         /* Ignore positive scores. */
         if (score_best >= 0.)
            continue;
 
         /* Add the lane. */
         presence_budget[fi][si] -= cost_best;
         if (presence_budget[fi][si] >= cost_cheapest_other)
            turns_next_time++;
         else {
            presence_budget[fi][si] = 0.; /* Nothing more to do here; tell ourselves. */
            if (lal[fi] == NULL)
               lal_bases[fi] -= sys_to_first_vertex[1+si] - sys_to_first_vertex[si];
         }
         safelanes_updateConductivity( ei_best );
         vertex_fmask[edge_stack[ei_best][0]] |= (MASK_1<<fi);
         vertex_fmask[edge_stack[ei_best][1]] |= (MASK_1<<fi);
         lane_faction[ ei_best ] = faction_stack[fi].id;
      }
   }
 
#if DEBUGGING
   for (int fi=0; fi<array_size(faction_stack); fi++)
      if (lal[fi] != NULL)
         assert( "Correctly tracked row offsets between the 'lal' and 'utilde' matrices" && lal[fi]->nrow == lal_bases[fi] );
#endif /* DEBUGGING */
 
   for (int fi=0; fi<array_size(faction_stack); fi++)
      cholmod_free_dense( &lal[fi], &C );
   free( lal );
   free( lal_bases );
   array_free( edgeind_opts );
   array_free( facind_vals );
   array_free( facind_opts );
 
   return turns_next_time;
}
 
static int cmp_key( const void* p1, const void* p2 )
{
   double d = cmp_key_ref[*(int*)p1] - cmp_key_ref[*(int*)p2];
   return SIGN( d );
}
 
static int safelanes_triangleTooFlat( const vec2* m, const vec2* n, const vec2* p, double lmn )
{
   const double MAX_COSINE = cos(MIN_ANGLE);
   double lnp = vec2_dist( n, p );
   double lmp = vec2_dist( m, p );
   double dpn = ((n->x-m->x)*(n->x-p->x) + (n->y-m->y)*(n->y-p->y)) / ( lmn * lnp );
   double dpm = ((m->x-n->x)*(m->x-p->x) + (m->y-n->y)*(m->y-p->y)) / ( lmn * lmp );
   return (dpn > MAX_COSINE && lnp < lmn) || (dpm > MAX_COSINE && lmp < lmn);
}
 
static int vertex_faction( int vi )
{
   const StarSystem *sys = system_getIndex(vertex_stack[vi].system);
   switch (vertex_stack[vi].type) {
      case VERTEX_SPOB:
         return sys->spobs[vertex_stack[vi].index]->presence.faction;
      case VERTEX_JUMP:
         return -1;
      default:
         ERR( _("Safe-lane vertex type is invalid.") );
   }
}
 
static const vec2* vertex_pos( int vi )
{
   const StarSystem *sys = system_getIndex(vertex_stack[vi].system);
   switch (vertex_stack[vi].type) {
      case VERTEX_SPOB:
         return &sys->spobs[vertex_stack[vi].index]->pos;
      case VERTEX_JUMP:
         return &sys->jumps[vertex_stack[vi].index].pos;
      default:
         ERR( _("Safe-lane vertex type is invalid.") );
   }
}
 
static inline int FACTION_ID_TO_INDEX( int id )
{
   for (int i=0; i<array_size(faction_stack) && faction_stack[i].id <= id; i++)
      if (faction_stack[i].id == id)
         return i;
   return -1;
}
 
static inline FactionMask MASK_ANY_FACTION()
{
   return ~MASK_0;
}
 
static inline FactionMask MASK_ONE_FACTION( int id )
{
   int ind = FACTION_ID_TO_INDEX( id );
   return ind>0 ? (MASK_1)<<ind : MASK_ANY_FACTION();
}
 
static inline FactionMask MASK_COMPROMISE( int id1, int id2 )
{
   FactionMask m1 = MASK_ONE_FACTION(id1), m2 = MASK_ONE_FACTION(id2);
   return (m1 & m2) ? (m1 & m2) : (m1 | m2);  /* Any/Any -> any, Any/f -> just f, f1/f2 -> either. */
}
 
static inline void triplet_entry( cholmod_triplet* m, int i, int j, double x )
{
   ((int*)m->i)[m->nnz] = i;
   ((int*)m->j)[m->nnz] = j;
   ((double*)m->x)[m->nnz] = x;
   m->nnz++;
}
 
static cholmod_dense* safelanes_sliceByPresence( cholmod_dense* m, double* sysPresence )
{
   size_t nr, nc, in_r, out_r;
   cholmod_dense *out;
 
   nr = 0;
   nc = m->ncol;
   for (int si=0; si < array_size(sys_to_first_vertex) - 1; si++)
      if (sysPresence[si] > 0)
         nr += sys_to_first_vertex[1+si] - sys_to_first_vertex[si];
 
   out = cholmod_allocate_dense( nr, nc, nr, CHOLMOD_REAL, &C );
 
   in_r = out_r = 0;
   for (int si=0; si < array_size(sys_to_first_vertex) - 1; si++) {
      int sz = sys_to_first_vertex[1+si] - sys_to_first_vertex[si];
      if (sysPresence[si] > 0) {
         for (size_t c = 0; c < nc; c++)
            memcpy( &((double*)out->x)[c*out->d + out_r], &((double*)m->x)[c*m->d + in_r], sz * sizeof(double) );
         out_r += sz;
      }
      in_r += sz;
   }
   return out;
}
 
static cholmod_dense* ncholmod_ddmult( cholmod_dense* A, int transA, cholmod_dense* B )
{
#if I_LOVE_FORTRAN
   blasint M = transA ? A->ncol : A->nrow, K = transA ? A->nrow : A->ncol, N = B->ncol, lda = A->d, ldb = B->d, ldc = M;
   assert( K == (blasint) B->nrow );
   cholmod_dense *out = cholmod_allocate_dense( M, N, M, CHOLMOD_REAL, &C );
   double alpha = 1., beta = 0.;
   BLASFUNC(dgemm)( transA ? "T" : "N", "N", &M, &N, &K, &alpha, A->x, &lda, B->x, &ldb, &beta, out->x, &ldc);
#else /* I_LOVE_FORTRAN */
   size_t M = transA ? A->ncol : A->nrow, K = transA ? A->nrow : A->ncol, N = B->ncol;
   assert( K == B->nrow );
   cholmod_dense *out = cholmod_allocate_dense( M, N, M, CHOLMOD_REAL, &C );
   cblas_dgemm( CblasColMajor, transA?CblasTrans:CblasNoTrans, CblasNoTrans, M, N, K, 1, A->x, A->d, B->x, B->d, 0, out->x, out->d);
#endif /* I_LOVE_FORTRAN */
   return out;
}
 
static double safelanes_row_dot_row( cholmod_dense* A, cholmod_dense* B, int i, int j )
{
   assert( A->ncol == B->ncol );
#if I_LOVE_FORTRAN
   blasint N = A->ncol, incA = A->d, incB = B->d;
   return BLASFUNC(ddot)( &N, (double*)A->x + i, &incA, (double*)B->x + j, &incB);
#else /* I_LOVE_FORTRAN */
   return cblas_ddot( A->ncol, (double*)A->x + i, A->d, (double*)B->x + j, B->d);
#endif /* I_LOVE_FORTRAN */
}